Smooth Torus Research Articles

Toric structures on near-symplectic 4-manifolds

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric structure generalizes the moment map image of a toric 4-manifold in the spirit of earlier work by the second author on almost toric symplectic 4-manifolds. We use the geometric structure on the base to investigate the problem of making given smooth torus actions on 4-manifolds symplectic or Hamiltonian with respect to near-symplectic structures and to give interesting constructions of structures which are locally given by torus actions but have nontrivial global monodromy.

GENERALIZED SPECIAL LAGRANGIAN TORUS FIBRATION FOR CALABI–YAU HYPERSURFACES IN TORIC VARIETIES I

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi–Yau hypersurfaces in toric varieties near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi–Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit X0. In this paper, we will deal with the region near the smooth top dimensional torus fibers of X0 and its mirror dual situation — the region near the 0-dimensional fibers of X0.

Interior crises in quasiperiodically forced period-doubling systems

As a representative model for quasiperiodically forced period-doubling systems, we consider the quasiperiodically forced logistic map, and investigate the dynamical mechanism for the interior crises. For small quasiperiodic forcing ε, a chaotic attractor abruptly widens via a “standard” interior crisis when it collides with a smooth unstable torus. However, as ε passes a threshold value, the smooth unstable torus loses its accessibility from the interior of the basin of the attractor. For this case, we use the rational approximation to the quasiperiodic forcing, and find that a nonstandard interior crisis occurs for a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor when it collides with an invariant “ring-shaped” unstable set. Particularly, we note that a three-band smooth torus transforms into a single-band intermittent strange nonchaotic attractor through the nonstandard interior crisis. The intermittent strange nonchaotic attractor is also characterized in terms of the average interburst time and the local Lyapunov exponent.

A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

Clumpy tori around active galactic nuclei

We discuss the question whether the matter in dusty tori around active galactic nuclei has a smooth or a clumpy structure. Nenkova et al. (2002, ApJ, 570, L9) have argued that the lack of emission feature in the SEDs of type 1 AGN galaxies combined with a clear absorption feature in type 2 AGN can be explained if the circumnuclear dust is distributed in discrete clumps. Our aim is to verify this. We use multi-dimensional radiative transfer models of smooth and clumpy tori, and compare the SEDs of equivalent smooth and clumpy models. We find that the 10 μm emission feature of the clumpy models, when seen almost face-on, is not appreciably reduced compared to the equivalent smooth models. Some of the clumpy models have a weak or even absent 10 μm feature, but so do some of the smooth models. On the whole the SEDs of clumpy and smooth tori are similar, but some details are different. The absorption feature seen at edge-on inclinations appears to be less deep in the clumpy models than in the smooth models, and the average flux in the near-infrared regime is stronger in the clumpy models. Moreover, at these inclinations the clumpy models have a slightly wider SED. Whether these differences are unique enough to be used as a diagnostic for clumpiness of AGN tori is not yet clear.

Smooth and nonsmooth dependence of Lyapunov vectors upon the angle variable on a torus in the context of torus-doubling transitions in the quasiperiodically forced Hénon map

A transition from a smooth torus to a chaotic attractor in quasiperiodically forced dissipative systems may occur after a finite number of torus-doubling bifurcations. In this paper we investigate the underlying bifurcational mechanism, which is responsible for the termination of the torus-doubling cascades on the routes to chaos in invertible maps under external quasiperiodic forcing. We consider the structure in the vicinity of a smooth attracting invariant curve (torus) in the quasiperiodically forced Hénon map and characterize it in terms of Lyapunov vectors, which determine the directions of contraction for an element of phase space in a vicinity of the torus. When the dependence of the Lyapunov vectors upon the angle variable on the torus is smooth, regular torus-doubling bifurcation takes place. On the other hand, we observe a transition consisting of the appearance of a nonsmooth dependence of the Lyapunov vectors upon the angle variable on the torus. We show that torus doubling becomes impossible after this transition has occurred, although the attractor of the system still remains a smooth torus. We illustrate how the transition terminates the torus-doubling bifurcation line in the parameter space with the torus transforming directly into a strange nonchaotic attractor. We argue that the transition plays a key role in mechanisms of the onset of chaos in quasiperiodically forced invertible dynamical systems.

Dynamical mechanism for band-merging transitions in quasiperiodically forced systems

As a representative model for quasiperiodically forced period-doubling systems, we consider the quasiperiodically forced logistic map, and investigate the mechanism for the band-merging transition. When the smooth unstable torus loses its accessibility from the interior of the basin of an attractor, it cannot induce the “standard” band-merging transition. For this case, we use the rational approximation to the quasiperiodic forcing and show that a new type of band-merging transition occurs for a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor through a collision with an invariant ring-shaped unstable set which has no counterpart in the unforced case. Particularly, a two-band smooth torus is found to transform into a single-band intermittent strange nonchaotic attractor via a new band-merging transition, which corresponds to a new mechanism for the appearance of strange nonchaotic attractors. Characterization of the intermittent strange nonchaotic attractor is made in terms of the average time between bursts and the local Lyapunov exponents.

Mechanism for boundary crises in quasiperiodically forced period-doubling systems

We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing ɛ, a chaotic attractor disappears suddenly via a “standard” boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of ɛ, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant “ring-shaped” unstable set which has no counterpart in the unforced case.

Fractalization route to strange nonchaotic dynamics

In the fractalization route for the formation of strange nonchaotic attractors (SNA's) in quasiperiodically driven nonlinear dynamical systems, a smooth torus gradually becomes a fractal as the forcing amplitude is increased, while the Lyapunov exponent remains nonpositive. Using techniques introduced by Kim et al. to identify unstable sets in SNA's, we study torus fractalization in a sequence of approximations wherein the quasiperiodic drive is replaced by periodic forcing of increasing period. This allows us to identify an unstable set embedded in the attractor. In the periodically forced system, we show that there is a cascade of attractor merging crises, and argue that the quasiperiodic analogue of such crises causes fractalization of tori to create SNA's.

Universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems

To examine the universality for the intermittent route to strange nonchaotic attractors (SNAs), we investigate the quasiperiodically forced Henon map, ring map and Toda oscillator which are high-dimensional invertible systems. In these invertible systems, dynamical transition to an intermittent SNA occurs via a phase-dependent saddle–node bifurcation, when a smooth torus collides with a 'ring-shaped' unstable set. We note that this bifurcation mechanism for the appearance of intermittent SNAs is the same as that found in a simple system of the quasiperiodically forced noninvertible logistic map. Hence, the intermittent route to SNAs seems to be 'universal', in the sense that it occurs through the same mechanism in typical quasiperiodically forced systems of different nature.

Intermittent Strange Nonchaotic Attractors in Quasiperiodically Forced Systems

Intermittent strange nonchaotic attractors (SNAs) appear typically in quasiperiodically forced systems. As a basic model, we consider the quasiperiodically forced Henon map and investigate the mechanism for the intermittent transition to SNAs. Using rational approximations to the quasiperiodic forcing, it is shown that dynamical transition to an intermittent SNA occurs via a phase-dependent saddle-node bifurcation when a smooth torus collides with a new kind of invariant “ring-shaped” unstable set.

SPECIAL LAGRANGIAN CYCLES AND HERMITIAN YANG–MILLS CONNECTIONS

We consider the adiabatic limit of a sequence of Hermitian Yang–Mills connections on a SU(r)-bundle over a semi-flat smooth special Lagrangian torus fibration π:M→B. The restriction of these connections to the fiber tori can be viewed as a family of SU(r)-connections on the fiber tori which are parameterized by B. We show that there is a gauge equivalent subsequence such that the restriction of each connection to the fiber tori converges in the Hausdorff topology, away from some closed subset of B of codimension at least two, to a limit which defines a r-sheeted special Lagrangian cycle in the dual special Lagrangian torus fibration.

Mechanism for the intermittent route to strange nonchaotic attractors.

Intermittent strange nonchaotic attractors (SNAs) appear typically in quasiperiodically forced period-doubling systems. As a representative model, we consider the quasiperiodically forced logistic map and investigate the mechanism for the intermittent route to SNAs using rational approximations to the quasiperiodic forcing. It is found that a smooth torus transforms into an intermittent SNA via a phase-dependent saddle-node bifurcation when it collides with a new type of "ring-shaped" unstable set.

Isotopy invariants for smooth tori in 4-manifolds

In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.

CHAOTIC BEHAVIOR OF THREE COMPETING SPECIES OF MAY–LEONARD MODEL UNDER SMALL PERIODIC PERTURBATIONS

The influence of periodic perturbations to a Lotka–Volterra system, modeling a competition between three species, is studied, provided that in the unperturbed case the system has a unique attractor — a contour of heteroclinic orbits joining unstable equilibria. It is shown that the perturbed system may manifest regular behavior corresponding to the existence of a smooth invariant torus, and, as well, may have chaotic regimes depending on some parameters. Theoretical results are confirmed by numerical simulations.

Quantum chaos on a toral phase space for general boundary conditions: recent new results

This paper is a brief review of recent new results for the quantum-chaos problem on a 2D toral phase space for general boundary conditions (BCs) on the torus. The following results will be reviewed and discussed: (a) The new concept of band distribution for Hamiltonians periodic in phase space. (b) Expressions for the general smooth torus map and for the condition determining the allowed quantum BCs associated with this map. (c) An exact renormalization scheme for quantum Anosov maps on the torus, which “eliminates” the general-BCs problem, replacing it by a fixed-BCs one.

Moment maps and non-compact cobordisms

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of the normal bundles of the connected components of the fixed points set. We use this to give simple new proofs of two formulas: Guillemin's topological version of the abelian Jeffrey-Kirwan localization, and the Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.

Handlebody decompositions of 4-manifolds and torus fibrations

In this theorem, a and b are the numbers of 2 and 3 handles, respectively. The type of the singular fiber in the above fibration is not necessarily 'good' in the sense of [7]. Theorem A was first announced in 1982 in [6]. (See also [7], [8].) The main reason for the long delay in publishing the proof is, of course, the author's laziness. But a reason was partly because the author was not fully convinced of the usefulness of the result; the variety of the singular fibers appearing in the construction seemed quite uncontrolable. As a matter of fact, such wide variety was a key to the proof of the existence theorem. Recently, the author received an enquiry from Daniel Ruberman about the proof. In trying to answer him, the author found a new example of a smooth torus fibration of S over S 2 applying the general construction in this paper to S. Also, he found that, if H2(M;Z) φ {0}, we can arrange so that the general fiber is not homologous to zero in M (Theorem B in Section 3). He hopes that these improvements might justify this late publication of the proof. The author thanks D. Ruberman, whose enquiry gave him an opportunity to publish this paper.

MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES

Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

Cohomology of regular embeddings

We give in this paper an explicit description of the rational cohomology ring of a “complete symmetric variety” or regular compactification of an algebraic symmetric variety. The motivation for this computation comes from the desire to describe an explicit Schubert calculus for these varieties, which have been considered since the work of Chasles and Schubert in several special cases. The technique we use is the result of our attempts to understand better the work of Jurkiewicz and Danilov in the case of torus embeddings. The combinatorial presentation of the cohomology ring of a smooth torus embedding finds a natural explanation and proof using the language of equivariant cohomology. The key point is that the Reisner-Stanley algebra of the torus embedding coincides with its equivariant cohomology ring. We consider a class of spaces, regular embeddings, that contains both “complete symmetric varieties” and smooth torus embeddings. By suitably generalizing the notion of Reisner-Stanley algebra, we are able to compute the rational equivariant cohomology ring of these spaces in an explicit way.